The Oilers are looking to move Gagner, among others, as they attempt to shake up their roster for the 2014-15 season.
In addition to Gagner, the Oilers are trying to move wingers Ales Hemsky and Ryan Smyth, and have fielded calls for Nail Yakupov and Jordan Eberle.
The Kings are trying to acquire some help for their special teams, as well as a goal-scoring left wing, and a top-four defenseman.
According to multiple reports, the Kings would ship left wing Kyle Clifford as part of a package for Gagner, who carries a $4.8 million salary cap hit.
The Oilers may also be picking up part of Gagner's salary, if this deal goes through, as the Kings have limited cap space.
A well-placed source, with knowledge of the situation, told TFP the Oilers have asked about Kings defenseman Jake Muzzin, but were told he isn't available. The Oilers have also spoken about Kings blueliner Alec Martinez, who could be part of a trade.
The Kings are also in talks with the New York Islanders about left wingers Thomas Vanek and Michael Graber, and defenseman Andrew MacDonald.
The Islanders are hoping to move Vanek by Friday's 3pm ET Olympic roster freeze, and if he's not dealt to the Kings, Los Angeles could focus on Grabner.
Kings GM Dean Lombardi has also expressed an interest in Buffalo Sabres left wing Matt Moulson and center Steve Ott.